@article{Zhang2023, 
author = {Rui Zhang and Yang Li and Xiaofei Yan},
title = {Exponential sums involving the divisor function over arithmetic progressions},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {5},
pages = {11084-11094},
keywords = {exponential sum, divisor function, arithmetic progressions},
url = {https://www.sciopen.com/article/10.3934/math.2023561},
doi = {10.3934/math.2023561},
abstract = {Let    ϕ  (  x  ) be a smooth function supported on    [  1  ,  2  ] with derivatives bounded by        ϕ          (      j      )        (  x  )  ≪  1 and        d    3    (  n  ) be the number of ways to write    n as a product of three factors. We get the asymptotic formula for the nonlinear exponential sum        ∑          n            ≡            l            m      o      d            q            d    3    (  n  )  ϕ      (          n      X        )    e      (                  3                              k            n                    3                    q        )  .}
}